Exporting a Blender image sequence animation to unity
You can't export modifiers through fbx, even less modifiers with animations. Model animation wise, fbx can only store skeletal animations and shape keys.

For this particular one, you have two real options:


Apply each frame as shape key (go to each frame individually, make copy of the modifier, then click "Apply as Shape Key". As result you should have key for each frame in "Properties > Data" tab. Fbx will export shape keys to unity (called "blend shapes" in unity), though you will need to use unity's animation tools and/or scripting to cycle through the blend shapes. There's also unity plugins for doing this.
Script that in unity. For that simple and seemingly random displacement script/shader works just as well if not better and more efficiently. (details of implementing this are not topic of this site but of Unity answers or Game development SE)